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1. Homework Statement [/b]
If z is a complex number, prove that there exists an r > 0 and a complex number w with |w|=1 such that z=rw. Are w and r always uniquely determined by z?
n/a
As C is a set closed under multiplication, we can define z=rw where r=(a,0) and w=(x,y). Hence z=(a,0)(x,y)=(ax,ay).
We can choose a w such that |w|=1; sqrt{(x+yi)(x-yi)}=1 = x^2+y^2=1 and clearly, this is the equation of the unit circle. Therefore, w must be determined uniquely by z so that they could intersect.
I don't know if my proof is valid and I'm aware of that I didn't say anything about r but I think it is related to z=(a,0)(x,y)=(ax,ay) somehow. Help please...
If z is a complex number, prove that there exists an r > 0 and a complex number w with |w|=1 such that z=rw. Are w and r always uniquely determined by z?
Homework Equations
n/a
The Attempt at a Solution
As C is a set closed under multiplication, we can define z=rw where r=(a,0) and w=(x,y). Hence z=(a,0)(x,y)=(ax,ay).
We can choose a w such that |w|=1; sqrt{(x+yi)(x-yi)}=1 = x^2+y^2=1 and clearly, this is the equation of the unit circle. Therefore, w must be determined uniquely by z so that they could intersect.
I don't know if my proof is valid and I'm aware of that I didn't say anything about r but I think it is related to z=(a,0)(x,y)=(ax,ay) somehow. Help please...